JEE Mathematics Statistics Notes Set 01

Measures of Central Tendency

An average is a single value that represents a whole distribution. Mathematical averages are the arithmetic, geometric and harmonic means; positional averages are the median and the mode.
Arithmetic mean: \( \bar x=\frac{\sum x_i}{n} \) for raw data and \( \bar x=\frac{\sum f_ix_i}{N} \) with \( N=\sum f_i \) for a frequency distribution.
Properties: \( \sum(x_i-\bar x)=0 \); \( \sum(x_i-\bar x)^2 \) is smaller than the sum of squares of deviations from any other number; if every value is replaced by \( ax_i+b \), the mean becomes \( a\bar x+b \).
For grouped data with class mark \( x_i \), assumed mean A and class width h, the step-deviation method gives \( \bar x=A+h\frac{\sum f_iu_i}{N} \) where \( u_i=\frac{x_i-A}{h} \).
Combined mean of two groups: \( \bar x=\frac{n_1\bar x_1+n_2\bar x_2}{n_1+n_2} \), which always lies between \( \bar x_1 \) and \( \bar x_2 \).
The arithmetic mean uses every observation but is pulled by extreme values. Geometric mean: \( (x_1x_2\cdots x_n)^{1/n} \); harmonic mean: \( \frac{n}{\sum 1/x_i} \); for positive unequal values \( \text{A.M.}>\text{G.M.}>\text{H.M.} \).

Median and Mode

Median: arrange the data in order. For n values the median is the \( \frac{n+1}{2} \)th value when n is odd, and the mean of the \( \frac n2 \)th and \( \left(\frac n2+1\right) \)th values when n is even. It is not affected by extreme values, so it is preferred for skewed data.
For a grouped distribution, locate the median class where the cumulative frequency first reaches \( \frac N2 \): \( \text{Median}=L+\frac{\frac N2-F}{f}\,h \), where L is its lower limit, F the cumulative frequency before it, f its frequency and h its width.
Mode: the value that occurs most often. For grouped data, \( \text{Mode}=L+\frac{f_1-f_0}{2f_1-f_0-f_2}\,h \) with \( f_1 \) the frequency of the modal class and \( f_0,f_2 \) the frequencies of the classes before and after it.
For moderately skewed data the empirical relation is \( \text{Mode}=3\,\text{Median}-2\,\text{Mean} \). Quartiles divide ordered data into four equal parts; \( Q_i \) is the \( \frac{i(N+1)}4 \)th term and the quartile deviation is \( \frac{Q_3-Q_1}2 \).

Measures of Dispersion

Range = largest value minus smallest value; coefficient of range \( =\frac{L-S}{L+S} \).
Mean deviation about a number a: \( \frac{\sum f_i|x_i-a|}{N} \). It is smallest when taken about the median. Coefficient of mean deviation = mean deviation divided by the average used.
Variance: \( \sigma^2=\frac{\sum f_i(x_i-\bar x)^2}{N}=\frac{\sum f_ix_i^2}{N}-\bar x^2 \); standard deviation \( \sigma=\sqrt{\sigma^2} \). For the first n natural numbers the mean is \( \frac{n+1}2 \) and the variance is \( \frac{n^2-1}{12} \).
Coefficient of variation \( =\frac\sigma{\bar x}\times100 \); of two series, the one with the smaller coefficient is more consistent.

Effect of Change of Origin and Scale

If \( y_i=ax_i+b \), then \( \bar y=a\bar x+b \), \( \sigma_y=|a|\sigma_x \), \( \sigma_y^2=a^2\sigma_x^2 \) and the mean deviation is multiplied by |a|. Adding a constant (change of origin) never changes the variance, the standard deviation or the mean deviation.
Combined variance of two groups with sizes \( n_1,n_2 \), means \( \bar x_1,\bar x_2 \) and variances \( \sigma_1^2,\sigma_2^2 \): \( \sigma^2=\frac{n_1\sigma_1^2+n_2\sigma_2^2}{n_1+n_2}+\frac{n_1n_2}{(n_1+n_2)^2}(\bar x_1-\bar x_2)^2 \).
Correcting a wrongly recorded value: adjust the total and the sum of squares, then recompute the mean and variance.

Worked Examples

Worked Example 1. Find the median of 4, 6, 9, 4, 2, 8, 10 and the mode of 2, 4, 6, 8, 8, 12, 17, 6, 8, 9.
Answer:
Ordered data: 2, 4, 4, 6, 8, 9, 10, seven values, so the median is the 4th value, 6.
In the second set 8 occurs three times (6 occurs twice), so the mode is 8.

Worked Example 2. Find the median and the mode of the distribution: class 0-10 (5), 10-20 (8), 20-30 (15), 30-40 (16), 40-50 (6).
Answer:
N = 50 and the cumulative frequencies are 5, 13, 28, 44, 50. Since \( \frac N2=25 \), the median class is 20-30: \( \text{Median}=20+\frac{25-13}{15}\times10=28 \).
The modal class is 30-40: \( \text{Mode}=30+\frac{16-15}{32-15-6}\times10=30+\frac{10}{11}\approx30.91 \).

Worked Example 3. The mean and variance of 7 observations are 8 and 16. Five of them are 2, 4, 10, 12, 14. Find the other two.
Answer:
Let them be x and y. Then \( x+y=56-42=14 \) and \( x^2+y^2=7(16+64)-460=100 \).
So \( xy=48 \) and the observations are 6 and 8.

Worked Example 4. Two groups have sizes 20 and 10, means 11 and 8, and variances 4 and 34. Find the combined variance.
Answer:
Combined mean \( =\frac{220+80}{30}=10 \).
\( \sigma^2=\frac{20(4)+10(34)}{30}+\frac{20\cdot10}{900}(11-8)^2=14+2=16 \).